Solution

The solution works in (what I presume is called) version 2 of the SOAP kit for Magento. I mean, I don’t even remember how I found out there is a “version 2” of this thing, I stumbled across it somewhere. Anyway, using my own trusty WSDL-Dumper Gimme Gimme Thingie, I found the version 2 counterparts for manipulating Tiered Pricing: catalogProductAttributeTierPriceInfo and catalogProductAttributeTierPriceUpdate. And here’s what I ended up to get Delete Tiered Pricing to work!

This code does delete Tiered Pricing, in 1.4.2.0 and 1.5.0.1

define('MAGENTO_WSDL_URL_V2',"http://www.myMagentoStore.com/api/v2_soap?wsdl=1");/* Delete Tiered Prices using V2 API: */
try {$cli2=new SoapClient(MAGENTO_WSDL_URL_V2);$sess2=$cli2->login(USER, PASS);/* Just grabbing current info and dumping to stdout,
* Important note: the 'sku' argument is a literal argument. It says that
* "value in $sku is the sku attribute for the product I'm looking for."
*/$tp=$cli2->catalogProductAttributeTierPriceInfo($sess2,$sku,'sku');print_r($tp);$killPrices[]=array('website'=>'all','customer_group_id'=>'all','qty'=>'','price'=>'');/* Killing the pricing by updating with "empty tiers" (insert joke here)
* Once again, the last argument 'sku' is a literal, describing the value
* I'm passing in $sku as an SKU. It could be, for example a product ID.
*/$rc=$cli2->catalogProductAttributeTierPriceUpdate($sess2,$sku,$killPrices,'sku');/* I'm printing the value returned in $rc, though I've never seen it be
* anything other than 1.
*/echo"\t----->".$rc."<-------\n";/* After the kill, this should come back empty: */$tp2=$cli2->catalogProductAttributeTierPriceInfo($sess2,$sku,'sku');print_r($tp2);}
catch( Exception $e){print("Tried killing tier price for $sku: ".$e->getMessage()."\n");print_r($e);}